Harmonic Progression(H.P)

Definition:
The sequence a1 , a2 , a3…….an……(ai ≠ 0) is said to be an H.P. if the sequence
$ \displaystyle \frac{1}{a_1} , \frac{1}{a_2} , \frac{1}{a_3} —- \frac{1}{a_n} $ is an A.P.

nth Term of H. P. :

The nth term an of the H.P. is

$ \displaystyle a_n = \frac{1}{a+(n-1)d} $

where ; $ \displaystyle a = \frac{1}{a_1} \, ; \, d = \frac{1}{a_2}-\frac{1}{a_1}$

Note:

# There is no formula for the sum of n terms of an H.P.

Harmonic Means:

∎ If a and b are two non-zero numbers, then the harmonic mean of a and b is a number H such that the numbers a, H, b are in H.P. We have

$ \displaystyle \frac{1}{H} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b}) $

$ \displaystyle H = \frac{2ab}{a+b} $

∎ If a1, a2, …….an are ‘ n ‘ non-zero numbers, then the harmonic mean H of these numbers is given by

$ \displaystyle \frac{1}{H} = \frac{1}{n}(\frac{1}{a_1}+ \frac{1}{a_2} +—- + \frac{1}{a_n}) $

∎ The n numbers H1, H2,…….,Hn are said to be ‘ n ‘ harmonic means between a and b, if a , H1 , H2…….., Hn , b are in H.P. i.e if are in A.P.

Let d be the common difference of the A.P., then

$ \displaystyle \frac{1}{b} = \frac{1}{a} + (n+1)d $

$ \displaystyle d = \frac{a-b}{(n+1)ab} $

Illustration : If a, b, c be in H.P. prove that:

$\large (\frac{1}{a} + \frac{1}{b} – \frac{1}{c}) (\frac{1}{b} + \frac{1}{c} – \frac{1}{a}) = \frac{4}{ac}  – \frac{3}{b^2} $

Solution: a, b, c are in H.P.

$\large \frac{2}{b} = \frac{1}{a} + \frac{1}{c}$

Now, $\large (\frac{1}{a} + \frac{1}{b} – \frac{1}{c}) (\frac{1}{b} + \frac{1}{c} – \frac{1}{a}) = (\frac{1}{b} + \frac{1}{a} – \frac{1}{c}) (\frac{1}{b} – (\frac{1}{a} – \frac{1}{c}) ) $  

$\large = (\frac{1}{b})^2 – (\frac{1}{a} – \frac{1}{c})^2$

$\large = (\frac{1}{b})^2 – [(\frac{1}{a} + \frac{1}{c})^2 – \frac{4}{a c}]$

$\large = (\frac{1}{b})^2 – [(\frac{2}{b})^2 – \frac{4}{a c}]$

$\large = \frac{4}{ac} – \frac{3}{b^2}$

Exercise :

(i) If H be the harmonic mean of a and b then find the value of

$\large \frac{H}{a} + \frac{H}{b}  – 2 $

(ii) If $\large \frac{b c}{a d} = \frac{b +c}{a +d} = \frac{3(b -c)}{a -d}$ ;  then show that a, b, c and d are in H. P.

(iii) Show that if a (b − c)x2 + b(c − a)xy + c(a − b)y2 is a perfect square, the quantities a, b, c are in harmonic progression.

(iv) If a , b , c be in A.P and a2 , b2 , c2 be in H.P, then prove that −a/2 , b , c are in G.P or else a = b = c.

Also Read :

→ Arithmetic Progression(G.P)
→ Geometric Progression(G.P)
→ Arithmetico Geometric Progression(G.P)
→ Miscellaneous Progression
→ INEQUALITIES
→ Solved Problems : Progression & Series

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